A related regularization idea is that of exploiting
decay of correlation and
approximate conditional independence in the filtering
distribution.
This structure is common when filtering spatiotemporal processes.
In this paper,
we introduced simple
localization
ideas
for nonlinear updates by imposing specific sparsity patterns in the transport maps composing the update.
In future work, one could explore
a much wider range of regularization techniques.
For instance, it is natural to consider
LASSO-type estimators in order
to “discover”—rather than to enforce—sparsity of the nonlinear transformations.
Or one might exploit low-rank structure in the change
from forecast to filtering distribution in order to reduce the effective dimension
of the nonlinear update, following
the line of work in [1]}, [2]}.
Initial accounts of low-rank structure in transport maps can be found in
[3]} and [4]}.
In general, we believe that practical high-dimensional filtering
relies crucially on the development of
structure-exploiting
regularization techniques.
